I am confronted with this definition which is unfamiliar to me:
Suppose $F\subset \mathbb C\times \mathbb C$ is a relation. $F$ is said to be algebraic if there exists a matrix $A\in M_{n+1}(\mathbb C)$ such that no row of $A$ is the zero vector, and
$$(x,y)\in F\iff \sum_{p=1}^{n+1}\sum_{q=1}^{n+1}A_{pq}y^{p-1}x^{q-1}=0$$
I'd like to find out more about it. I've been trying to find this in other texts without much luck, and hampered by the obvious problem of only having "algebraic relation" as a search term.
This appeared in the context of a text on analytic functions, at the end of a chapter about analytic relations, analytic continuation, and branch-points.
It might be useful to provide the remaining context provided by the book (there isn't much!) The definition above is only used for three following exercises, and the book does not go any further into this thing. The exercises are:
- If $F$ is an algebraic relation, show $F^{-1}$ is algebraic also.
- If $g$ is an analytic function and $g\subseteq F$, then $F$ contains every analytic relation extending $g$.
- If $G$ is an analytic relation contained in an algebraic relation, then $G'$ (the derivative of $G$) is also contained in an algebraic relation.
That's it... I think that's all that is mentioned.
Can someone point me to other texts where this idea appears and/or give other terminology that might lead me to see more about this?